Cycle Exclusion in the Collatz Iteration via an Elementary Correction-Sum Bound

For the Collatz map, every candidate cycle with a odd steps and b halvingsdetermines a branch fixed point x* = C(v)/(2b − 3a), where C(v) is the correctionsum of the halving pattern v = (v1, …, va). We prove a closed-form sharp boundCmax(a,b) = 3a−1 + (3a−1 − 2a−1)·2b−a+1 (Theorem 1) and deduce twoconsequences. First, non-trivial cycles with a ≤ 2 are impossible by size alone.Second, combining the bound with the computational verification of convergencefor all n < 268 (Barina 2020) excludes all non-trivial cycles with a ≤ 110 odd steps(Theorem 2); the argument is elementary and uses no bounds on linear forms inlogarithms. An ultrametric rigidity lemma shows that integrality of the branchfixed point automatically forces the orbit to realize its defining valuation pattern;the cycle problem is therefore exactly the divisibility problem, with no separateconsistency condition. We further prove a balance identity showing that, pernumber of odd steps, first-order equidistribution heuristics predict approximatelyone integer branch fixed point — and that this prediction is exactly saturated bythe trivial cycle. Non-trivial cycles are therefore a second-order phenomenon,invisible to counting and size arguments alike, consistent with thenon-Presburger-definability of the underlying divisibility predicate (Dhiman &Pandey 2026)